Volunteers who had developed a cold within the previous 24 hours were randomized to take either zinc or placebo lozenges every 2 to 3 hours until their cold symptoms were gone. Twenty-five participants took zinc lozenges, and 23 participants took placebo lozenges. The mean overall duration of symptoms for the zinc lozenge group was x1 = 4.4 days, and the standard deviation of overall duration of symptoms was 1.6 days. For the placebo group, the mean overall duration of symptoms was x2 = 8.1 days, and the standard deviation was 1.5 days. Compute a 95% confidence interval for the difference in mean days of overall symptoms for the placebo and zinc lozenge treatments. Use the unpooled standard error and use the smaller of n1 - 1 and n2 - 1 as a conservative estimate of degrees of freedom. (Round the answers to one decimal place.)
3 to 4 days
yes
To compute the 95% confidence interval for the difference in mean days of overall symptoms between the placebo and zinc lozenge treatments, we need to follow these steps:
Step 1: Calculate the standard error.
The standard error measures the precision of the estimated difference in means. In this case, we will use the unpooled standard error formula.
Standard Error (SE) = sqrt[(s1^2/n1) + (s2^2/n2)]
where s1 is the standard deviation for the zinc lozenge group (1.6 days), n1 is the number of participants in the zinc lozenge group (25), s2 is the standard deviation for the placebo group (1.5 days), and n2 is the number of participants in the placebo group (23).
SE = sqrt[(1.6^2/25) + (1.5^2/23)]
SE = sqrt[0.1024 + 0.0978]
SE = sqrt(0.2002)
SE ≈ 0.447
Step 2: Calculate the t-score.
The t-score is used to determine the critical value for the confidence interval. We will use a 95% confidence level, which corresponds to a significance level of α = 0.05. Given that we need to use the smaller of n1 - 1 and n2 - 1 as a conservative estimate of degrees of freedom, we will use degrees of freedom = min(25-1, 23-1) = 22.
Using a t-table or statistical software, find the t-value corresponding to a 95% confidence level with 22 degrees of freedom. The t-value is approximately 2.074.
Step 3: Calculate the margin of error.
The margin of error represents the range of values within which we expect the true mean difference to fall. It is calculated by multiplying the t-score by the standard error.
Margin of Error = t * SE
Margin of Error = 2.074 * 0.447
Margin of Error ≈ 0.925
Step 4: Calculate the confidence interval.
Finally, we can calculate the confidence interval by subtracting and adding the margin of error to the difference in mean days of overall symptoms between the placebo and zinc lozenge treatments.
Confidence Interval = (x2 - x1) ± Margin of Error
Confidence Interval = (8.1 - 4.4) ± 0.925
Confidence Interval = 3.7 ± 0.925
Rounded to one decimal place, the 95% confidence interval for the difference in mean days of overall symptoms between the placebo and zinc lozenge treatments is approximately 3.7 to 4.6 days.